u. S. ARMY RESEARCH 0FFICE Report No. 76-1 February 1976 TRANSACTIONS OF THE TWENTY-FIRST CONFERENCE OF ARMY MATHEMATICIANS Sponsored by the Army Mathematics Steering Committee Host U. S. Army White Sands Missile Range White Sands Missile Range, New Mexico 14-16 May 1975 •.• I Approved for public release; distribution unlimited. The findings in this report are not to be construed as an official Department of the Army position, un less so designated by other authorized documents. U. S. Army Research Office P. O. Box 12211 Research Triangle Park, North Carolina · I FOREWORD These Transactions preserve in print most of the invited addresses and contributed papers of the Twenty-first Conference of Army Mathemati cians. These meetings are sponsored by the Army Mathematics Steering Committee (AMSC) on behalf of the Office of the Chief of Research, Develop ment and Acquisition. Members of this Committee request that the guest lecterers be effective researchers who are in frontier fields of live current interest. They feel that the addresses by the invited speakers as well as contributed papers by Army per~onnel will stimulate the inter changes of ideas among the scientists attending said meetings. In June of 1973, Messrs. W. A. McCool and William Sheperd of the White Sands Missile Range called the Army Research Office (ARO) to inquire about the possibility of holding either the Twentieth or the Twenty-first Conference of Army Mathematicians at their installation. Since the next, the Twentieth Conference, had been scheduled for the US Army Natick Labor aotires, these gentlemen were thanked and advised that a written offer to host the 1975 meeting would be appreciated. Dr. Richard H. Duncan, Tech nical Director and Chief Scientist of the White Sands Missile Range, in a letter dated 10 May 1974 issued a formal invitation to hold this meeting at his installation. Part of his letter is quoted below: I cordially invite the Applied Mathematics Subcommittee, US Army Mathematics Steering Committee, to hold the Twenty-First Confer ence of Army Mathematicians at the US Army White Sands Missile Range, NM, in May 1975. We would be pleased to serve as host to the conference. iii I believe a conference held here would attract a number of Western Army Mathematicians who do not normally attend the conferences. Their attendance and participation would be of substantial benefit to the Army installations in our area. If the conference is held here, Mr. P. J. Higgins, Chief of the Analysis and Computation Division, National Range Operations Directorate, will be in charge of local arrangements. Please contact him for additional information. Over 115 Army and academic scientists attended this 14-16 May 1975 conference, which convened in E1 Paso, Texas. It is pleasing to note that over hald of these individuals were from the host installation. On the afternoon of the second day of the meeting, Mr. P. J. Higgins, the Chairman on Local Arrangments, arranged a trip to the White Sands Missile Range and a very interesting tour of that base. Those in attendance are indebted to him and members of his staff for this three day meeting. The success of this conference was due to many persons, including the active and enthusiastic members of the audience, the chairmen of the sessions, and authors of the many papers. The members of the AMSC would like to thank these gent1ement for taking time to prepare papers for these Transactions so that many persons unable to attend this symposium can profit by their contributions to the scientific literature. ;v TABLE OF CONTENTS* TITLE PAGE . Foreword .. .. .. .. iii Table of Contents v Program . . . . . viii On spectrally Bounded Signed Graphs A. J. Hoffman . 1 • • f • • '" • • • • • ill • Effects of Concentrated Mass and Thrust Directional Control on the Stability of Free-Free Beams Julian J. Wu ...•... . . . . . . . . . . . . . . . 7 The Standard Linear Model in the Stability and Mass Optimization of ~onconservative Euler Beams Charles R. Thomas . . • . . . . . . . . . . . . . . . . . . 27 Buckling of Orthotropic Rectangular Cylinders Earl C. Steeves . 79 Energy Release Rate in Terms of Complex Analytic Functions M. A. Hussain and S. L. Pu . . . . . . . . . . . . . 81 Development and Application of Dynamic r~athematical Models for Evaluation of Military Systems, Forces and Doctrine Roger F. Wi11is. . . . . . . . . . . . . . . . . . 93 Homeostatic Criteria for Assessment of the Morphological State of Large Scale Hybrid Analytical-Simulation Models Howard M. Bratt. . . . . . . . . . . . . . . . . 117 The Application of Infinitesimal Transformation Groups to the Solution of Nonlinear Partial Differential Equations George W. Ullrich. . . . . . . . . . . . . . • . . . . . 125 On Uniqueness of Piecewise Polynomial Approximation C. K. Chui, P. W. Smith, and J. D. Ward .... 141 Linear Generalizations of the Gronwall-Reid-Bellman Lemma Jagdish Chandra and Paul Wi11iam Davis . . . . . 149 *Th;s table of contents contains only the papers that are published in this technical manual. For a list of all papers presented at the Twenty-first Conference of Army Mathematicians, see the program of the agenda. v AMethod for the Numerical Solution of Two-Point Boundary Value Problems based on the Use of Volterra Integral Operations Ho Fujita and Lo Bo Rall 151 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Nonlinear Vibration Theory of Pavements . . . . . . Richard Ao Weiss .. ... .. .. .. .. 187 0 0 0 0 0 • 0 0 • Application of the Theory of Slender Curved Rods to the Analysis of Elastic Yarns No Co Huang • 217 0 • 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 •• 0 • AMaximum Likelihood Decision Algorithm for Markov Sequences with Multiple Applications to Digital Communications Andrew J. Viterbi . 251 0 0 0 • 0 0 0 0 0 0 The Calculation of Unsteady Shear Stresses in Gun Barrels Ro Yalamanchili 277 0000000000000000 Nonlinear Problems in the Interaction of a Steel Cartridge Case and the Chamber Jo Jo Toal and So Co Chu 299 0 Nine-Point Difference Solutions for Poisson's Equation J Barkley Ross . 313 0 0 • 0 0 • 0 0 0 0 0 0 0 0 0 III .. • III • • • Illuminating Round Effectiveness Modeling Martin Messinger and Leonard Oleniczak 335 Launching of Electromagnetic Surface Waves at a Planar Metal-Air Interface J. M. Zavada and Eo Lo Church • 351 0 0 0 0 • 0 ••• 0 • Comparison of Perturbation-Theoretic and Exact Calculations of Nonlinear Optical Properties of Optoelectronic Materials So So Mitra, L. Mo Narducci, and Ro Ao Shatas •• 363 0 • ATime-Dependent Quantized Natural Collision Coordinates Method Norman Mo Witriol 397 0 0 0 0 0 ••• 0 • 0 0 •• 0 •• 0 0 0 Computer Simulation of the Intermediate Ballistic Environment of a Small Arm Csaba Ko Zoltani 399 0 0 0 0 0 • 0 0 • 0 0 0 0 0 0 0 • 0 0 0 • 0 • 0 Generalized Shock Wave Physics: Electromagnetic and Second Sound Shocks Paul Harris 415 0 0 0 0 0 •• On Riemann's Invariant and Schock Impedance of Solids Yo Ko Huang 423 0 0 0 0 0 0 0 0 0 0 0 • 0 0 0 • 0 0 o 0 vi Frequency Dependent Wave Arrival Time Delays in Dispersive and Nondispersive Media J. R. Stabler, E. A. Baylot, and D. H. Cress. . . . . 431 The Method of Parabolic Substitution for High Subsonic Flow Klaus Oswatitsch and Robert E. Singleton .....•. 445 The Backward Beam Equation and the Numerical Computation of Dissipative Equations Backwards in Time Alfred Carasso ..•.......•••••••••• 457 Special Solutions of the One-Dimensional Parabolic Equation Siegfried H. Lehnigk . . . . . . . . . . . . . . . . . . . . . . 503 r Integration of ~ F{x)J (ax)Jl{bx)dx o 0 Shunsuke Takagi 511 Singular Perturbations in Heat Conduction and Diffusion Problems John F. Polk . . . . • • . . . . • . . . • • . . • . • . • . 543 Sample Sizes for Missile in Flight Reliability Determination Edward F. Southworth . . . . . . . . . . . . . . . . . . 559 DOVAP Best Estimate of Trajectory . . . Robert H. Turner and William S. Agee .. .' .. . ,. .. . ,. " . .. 587 Optimal DOVAP Instrumentation Planning William S. Agee and Jerry L. Meyer. 605 • ,. til • .. • • .. 11II • .' • • • Proving Programs Correct Elwood D. Baas • . . . • . . 615 .. • • • • .. 11II • • 11II • • • • .. • • • Generalized Plane Strain in an Elastic, Perfectly Plastic Cylinder, With Reference to the Hydrau1ic Autofrettage ProcesS Alexander S. Elder, Robert C. Tompkins and Thomas L. Mann 623 Nonlinear Problems in Chemically Reacting Diffusive Systems Donald S. Cohen ......•.......••..• 661 ANew Numerical Method of Solution of Schrodinger's Equation George Morales and Robert G. McIntyre . • . . . • 671 List of Attendees . . . . . • . . . • . . . . . . . . . . . . 707 vii j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j PROGRN1 21st CONFERENCE OF ARMY MATHEMATICIANS US Army White Sands ~1i ssile Range, NevI t1exico All general and technical sessions will be held in the Rodeway Inn (Bassett Center) 6201 Gateway West. El Paso, Texas. REGISTRATION - ERICA ROOM 0845--0900 OPENING OF THE CONFERENCE WELCOMING REMARKS - ERICA ROOM 0900-1000 GENERAL SESSION I - ERICA ROOM SPEAKER: Professor A1drl J. Hofflnan Mathematical Sciences IBM Thomas J. Watson Research Center P.O. Box 218 Yorktown Heights, New York 10598 TITLE: Graph Theory and Eigenvalues of Mcttrices CHAIRr\1AN: Dr. Ivan R. Hershner, Jr. Room 3E 453 The Pentagon HQDA (DAt-1A-ARZ-D) Washington, D. C. 20310 1000··'020 BREAK 1020-1200 TECHNIC.t\L SESSION I - ERICA ROor·l CHAIRr~AN : Dr. Edl-/al'd \·1. Ross, .Jr. Staff Mathematician US Arn~ Natick Laboratories Natick, Massachusetts 01760 viii Hednesday AM 1020-1200 TECHNICAL SESSION I (Continued) EFFECTS OF CONCENTRATED MASS AND THRUST DIRECTIONAL CONTROL ON THE STABILITY OF FREE-FREE BEAMS Julian J. Wu. B~n~t Weapons Laboratory. Watervliet Arsenal. Watervliet. New York THE STANDARD LINEAR MODEL IN THE STABILITY AND MASS OPTI~lI ZATION OF NONCONSERV,UIVE EULER BEAr·1S Charles R. Thomas. B~n~t \'Jeapons LaboratoY'y, Water'll iet Arsenal. Watervliet, New York BUCKLING OF ORTHOTROPIC RECTANGULAR CYLINDERS Earl C. Steeves, US Army Natick Laboratories. Natick. Massachusetts ENERGY RELEASE RATE IN TERMS OF COMPLEX ANALYTIC FUNCTIONS M. A. Hussain and S. L. Pu, Watervliet Arsenal~ Watervliet, New York 1020-1200 TECHNICAL SESSION II - TROPHY ROOM CHAIRMAN: Dr. Badrig M. Kurkjian At-1CRD-R 5001 Eisenhower Avenue Alexandria. Virginia 22304 DEVELOPMENT AND APPLICATION OF DYNA~lI Cr·1ATHH1ATICAL MODELS FOR EVALUATION OF MILITARY SYSTEMS, FORCES AND DOCTRINE Roger F. ~;i 11is, US Army Combined Arms Combat Development Activity, Ft. Leavem-lOrth, Kansas APPLIED TECHNIQUES OF ACQUISITION AND PRODUCTION COST ANALYSIS Van A. Puryear. US Army Troop Support Command. St. Louis, Missouri HOMEOSTATIC CRITERIA FOR ASSESSMENT OF THE MORPHOLOGICAL STATE OF LARGE SCALE HYBRID ANALOG-SInULATION Cor·1PUTER MODELS HOI'/ard ~t Bratt. US Army Air r~obi1ity Research and Development Laboratory, Ft. Eustis. Virginia ix